Lauren, 36, was seen with her seven-year-old son Adam at the family home in Bridgehampton, New York
Wearing a loose white shirt and cropped trousers, she already appeared to be showing a hint of a bump
Lauren's outing came as friends of Cowell insisted he will do everything he can to support the baby
By DANIEL BATES, SARA NATHAN and SARAH BULL
Lauren Silverman managed a smile as she broke her cover and appeared in public for the first time since it was revealed she is pregnant with Simon Cowell's baby.
The 36-year-old socialite was seen walking her seven-year-old son Adam down the driveway of the $4million family home in Bridgehampton, New York, where she has been staying since the baby news broke.
Lauren's outing came as a friend of music mogul Cowell, who is worth an estimated £200million, insisted he will do everything he can to support and provide for his child.
Lauren's baby news
The friend told MailOnline: 'Simon has given all his exes houses - so of course he's going to be generous to his own child', adding that 'he has not even considered not looking after this baby.
'Anyone who knows him knows that he's generous to a fault. Lauren keeps saying that she just wants to be with him, she's so in love with him, they're very good together. She makes him relax and have fun.
'She's fun loving, she loves a party and a giggle. Simon's himself around her, she's been good for him. He's so happy.'
Wearing a white shirt, black cropped leggings and a baseball cap, Lauren already appeared to be displaying a hint of a bump as she stepped out with her son.
Lauren was seen waving to a female friend as she was met halfway down the driveway, and smiled widely as she placed her hand on her son's back.
She then turned round and walked quickly back into the house as the friend ensured Adam was safely on the school bus.
Lauren has been staying at the seven-bedroom property with her mother Carole and another friend.
The house had been listed for rent recently and the description on a real estate website says it has four fireplaces, seven and a half bathrooms and a country eat-in kitchen.
Despite Lauren and Adam's relatively good spirits as they were pictured, it is likely that the child will be the main issue in his parents' divorce case.
Lauren's estranged husband Andrew Silverman made no comment about the baby news as he left his New York apartment on Wednesday evening
Andrew filed papers in the Manhattan Supreme Court two weeks ago, naming Cowell as a co-respondent and citing adultery as the reason for ending the marriage.
A source closed to Cowell told MailOnline that Andrew named him in the divorce ‘to cause the utmost embarrassment’ to the star.
The source said: ‘Andrew wants to embarrass Simon. There’s no way this will actually go to trial, but Andrew just wants to get back at him.’
Lauren's pregnancy news
And noted divorce lawyer Vanessa Lloyd Platt told MailOnline of Andrew's decision: 'It's a very unusual thing to name him. And considering they are all supposed to be friends it's a very unusual step to take.
'Andrew would have not only had to name him, but Simon would also have had to be served with the papers. It's all going to very muddy, muddled and an absolute nightmare for Simon.'
Lauren and her husband Andrew - seen here with the mogul in St Barts in January 2012
However, Vanessa said that she thinks reports that Cowell could be called to the witness stand to reveal details of the alleged affair is unlikely.
But it does seem more likely that he will have to reveal details of his personal wealth.
Vanessa added: 'It could be that in the fight between Lauren and Andrew, the financial circumstances debate would have to bring in Simon.
'Andrew's team are likely to be saying that when you look at Lauren's needs, you have to take into account that Simon will be looking after her.'
Lauren is said to be thrilled about the fact that she is pregnant with Simon's baby, with a source telling People that she is 'ecstatic' about her relationship with the mogul.
Not everyone is happy about the pregnancy news, however, as Andrew's brother Alexander Silverman insisted that the baby constituted a 'betrayal' of trust, because Simon and Andrew were close friends for many years.
He told the New York Post: 'It is an unbelievable story of betrayal. It is a sad story and a tragic story.'
No evidence has been revealed so far to suggest that Cowell and Lauren's affair started before the marriage began to fall apart.
Cowell maintained a stony silence on the subject as he was pictured for the first time since the pregnancy revelations while leaving a television studios in Los Angeles after filming The X Factor USA on Wednesday night.